Adjacency preservers on invertible hermitian matrices II

Maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are characterized. It is shown that such maps form a group that is generated by the maps A↦PAP⁎A↦PAP⁎, A↦AσA↦Aσ, and A↦A−1A↦A−1, where P   is an invertible matrix, P⁎P⁎ is its conjugate transpose, and σ is an automorphism of the underlying field. Bijectivity of maps is not an assumption but a conclusion. Moreover, adjacency is assumed to be preserved in one direction only.